T2 inversions with reduced motion artifacts

ABSTRACT

A method for processing nuclear magnetic resonance (NMR) measurement data includes: receiving, with a processor, NMR measurement data obtained from an NMR tool, the NMR measurement data being affected by a motion artifact and having a first echo train obtained with a long polarization time TW ET  and a second echo train obtained with a short polarization time TW TL  that is shorter than TW ET ; and at least one of (i) reducing, with a processor, an effect on the NMR measurement data of the motion artifact using the first echo train and the second echo train and (ii) identifying, with a processor, the motion artifact using the first echo train and the second echo train; wherein the motion artifact is related to a magnetic field magnitude that varies in a volume of interest due to a motion of the NMR tool.

BACKGROUND

Earth formations, or simply formations, may be used for various purposessuch as hydrocarbon production, geothermal production, and carbondioxide sequestration. In order to make optimal use of a formation, itis typically characterized using a downhole tool that is conveyedthrough a borehole penetrating the formation.

One type of downhole tool is a nuclear magnetic resonance (NMR) toolthat performs NMR measurements on the formation to determine variousproperties such as porosity for example. In one application referred toas logging-while-drilling, the NMR tool is coupled to a drill string.The NMR tool performs NMR measurements while the drill string isrotating causing a drill bit also coupled to the drill string to drillthe borehole. The drill process, however, may cause the drill string tomove laterally in the borehole thus continuously varying the distancefrom the NMR tool to the formation being characterized. Lateral motionof the NMR tool may also occur due to rotation of the tool withoutdrilling. Unfortunately, the continuously varying distance may inducemotion artifacts in the obtained NMR data resulting in a decrease in theaccuracy of the data.

BRIEF SUMMARY

Disclosed is a method for processing nuclear magnetic resonance (NMR)measurement data. The method includes: receiving, with a processor, NMRmeasurement data obtained from an NMR tool, the NMR measurement databeing affected by a motion artifact and having a first echo trainobtained with a long polarization time TW_(ET) and a second echo trainobtained with a short polarization time TW_(TL) that is shorter thanTW_(ET); and at least one of (i) reducing, with a processor, an effecton the NMR measurement data of the motion artifact using the first echotrain and the second echo train and (ii) identifying, with a processor,the motion artifact using the first echo train and the second echotrain; wherein the motion artifact is related to a magnetic fieldmagnitude that varies in a volume of interest due to a motion of the NMRtool.

Also disclosed is a method for performing nuclear magnetic resonance(NMR) measurements on an earth formation. The method includes: conveyingan NMR tool through a borehole penetrating the earth formation;receiving, with a processor, NMR measurement data obtained from the NMRtool disposed on the carrier, the NMR measurement data being affected bya motion artifact and having a first echo train obtained with a longpolarization time TW_(ET) and a second echo train obtained with a shortpolarization time TW_(TL) that is shorter than TW_(ET); and at least oneof (i) reducing, with a processor, an effect on the NMR measurement dataof the motion artifact using the first echo train and the second echotrain and (ii) identifying, with a processor, the motion artifact usingthe first echo train and the second echo train; wherein the motionartifact is related to a magnetic field magnitude that varies in avolume of interest due to a motion of the NMR tool.

BRIEF DESCRIPTION OF THE DRAWINGS

The following descriptions should not be considered limiting in any way.With reference to the accompanying drawings, like elements are numberedalike:

FIG. 1 illustrates a cross-sectional view of an exemplary embodiment ofa nuclear magnetic resonance (NMR) tool disposed in a boreholepenetrating the earth;

FIG. 2 depicts aspects of idealized NMR echo trains and idealizedtrainlets for illustration of an NMR motion artifact;

FIGS. 3A-3C, collectively referred to as FIG. 3, depict aspects an NMRecho train and trainlet used to demonstrate prior art joint inversion inthe presence of a motion artifact;

FIG. 4 depicts aspects of the T₂ distribution resulting from the priorart joint inversion;

FIGS. 5A-5C, collectively referred to as FIG. 5, depict aspects of anNMR echo train and trainlet used to demonstrate joint inversion withmotion correction;

FIG. 6 depicts aspects of the T₂ distribution resulting from the jointinversion with motion correction;

FIG. 7 depicts aspects of an NMR long echo train and trainlet with ashort wait time of 60 ms used to demonstrate prior art joint inversion;

FIG. 8 depicts aspects of the T₂ distribution resulting from the priorart joint inversion of the NMR long echo train with long wait time andtrainlet with the short wait time of 60 ms;

FIG. 9 depicts aspects of an NMR long echo train with long wait time andtrainlet with a short wait time of 60 ms used to demonstrate jointinversion with motion correction;

FIG. 10 depicts aspects of the T₂ distribution resulting from the jointinversion with motion correction of the NMR long echo train with longwait time and trainlet with the short wait time of 60 ms;

FIG. 11 depicts aspects of an NMR long echo train with long wait timeand two trainlet types with different wait times (one second and 60 ms)used to demonstrate joint inversion with motion correction;

FIG. 12 depicts aspects of the T₂ distribution resulting from the jointinversion with motion correction of the NMR long echo train with longwait time and two trainlet types with different wait times (one secondand 60 ms);

FIG. 13 presets tabled results of numerical examples; and

FIG. 14 is a flow chart for a method for estimating a property of anearth formation using NMR measurement data.

DETAILED DESCRIPTION

A detailed description of one or more embodiments of the disclosedapparatus and method presented herein by way of exemplification and notlimitation with reference to the figures.

Disclosed are method and apparatus for processing measurements performedby a nuclear magnetic resonance (NMR) tool that may be subject to motionin a borehole. Alternatively or in combination with tool motion, the NMRtool may have a non-axially symmetric magnetic field such that when thetool is rotated, the magnetic field magnitude varies at a fixed locationin the formation. The motion may cause the NMR measurements toinaccurately quantify properties of the formation. The inaccuracyinduced into the NMR measurements due to the motion (or non-axiallysymmetric magnetic field motion) is called a motion artifact. Theprocessing techniques disclosed herein identify a motion artifact andremove it from the NMR measurement data to provide corrected NMRmeasurement data that more accurately quantify properties of theformation.

Next apparatus for implementing the teachings herein is discussed. FIG.1 illustrates a cross-sectional view of an exemplary embodiment of anNMR tool 10 disposed in a borehole 2 penetrating the earth 3, whichincludes an earth formation 4. The NMR tool 10 is configured to performNMR measurements on the formation 4. The NMR measurements includegenerating NMR signal echoes from atomic nuclei, such as hydrogennuclei, of the formation. Long echo trains can be recorded. The decaysof the echo trains are caused by the so-called T₂ relaxation, also knownas transverse or spin-spin relaxation. The NMR measurements yieldtransverse relaxation times T₂, which are exponential decay timeconstants that correspond to a characteristic or property of theformation 4 material. Transverse relaxation relates to the loss of phasecoherence of the protons in the formation 4 material while precessingabout a static magnetic field during an NMR measurement. There is notone single value of T₂ for formation fluids but a wide distribution ofvalues lying anywhere between fractions of a millisecond and severalseconds for example. The quantitative distribution of T₂ values is theprincipal output of the NMR tool 10. A sequence of T2 distributionplotted versus depth in the borehole may be referred to as an NMR T2distribution log. The NMR tool 10 may also output longitudinalrelaxation time constants (T₁) associated with polarizing the nuclei inthe formation.

The T₂ (also referred to as T2) decay may be approximated by a sum ofexponential functions (multi-exponential approximation) resulting in aT2 distribution. The process of obtaining this T2 distribution iscommonly called T2 inversion, echo fit or mapping. From the T2distribution, total porosity, partial porosities, pore size and fluidtype in the formation may be determined—properties that are ofparticular interest. The long T2 components are usually called FreeFluid (FF) components or Bulk Volume Moveable (BVM); the medium T2components are usually called Bound Water (BW) components or Bulk VolumeIrreducible (BVI); and the short T2 components are usually called ClayBound Water (CBW) components.

The basic NMR method for obtaining the T2 echo decay is a long wait time(TW_(long), polarization time) to get close to (or achieve full)equilibrium polarization, followed by a train of several hundred toseveral thousand NMR echoes generated, for example, by a pulse echosequence such as the well-known Carr Purcell Meiboom Gill (CPMG) pulseecho sequence. The equilibrium polarization is useful to get the totalporosity from the start amplitude of the echo train. In addition to thelong echo train with long TW, echo trains with short TW_(short) (calledtrainlets or bursts) are used for a more accurate determination of theshort T2 components in the T2 distribution. Usually, but notnecessarily, the trainlets are shorter (i.e., they have a smaller numberof echoes) than the long echo train with long TW_(long).

In one or more embodiments, a number of NMR echo trains (with long TW)are acquired and averaged. Preferably TW being long enough to polarizeall NMR components fully. A number of NMR trainlets (with short TW) areacquired and averaged. Usually the trainlets have a small number ofechoes only (to save time, memory and power) and use a short TW (in theprior art: to save time and consequently increase signal-to-noise ratio(SNR) of the measurement; in the present disclosure to reduce motionartifacts). The trainlet TW should be long enough to polarize fully theT2 components that are later, after inversion, extracted from thetrainlet T2 distribution. The number of averaged trainlets is greaterthan the number of averaged echo trains with long TW in order to get abetter determination of the short T2 components. These acquired data maybe processed (i.e., inverted) according to an inversion method such asone of those discussed below.

Components in the NMR tool 10 includes a static magnetic field source 13that magnetizes formation materials and an antenna 14 that transmitsprecisely timed bursts of radio-frequency energy that provides anoscillating magnetic field. In a time period between these pulses, theantenna receives a decaying echo signal from those hydrogen protons thatare in resonance with the static magnetic field produced by the staticmagnetic field source at the transmitted RF frequency. NMR measurementsare performed in a toroidal volume surrounding the NMR tool 10 referredto as a volume of interest 9. Because a linear relationship existsbetween the proton resonance frequency and the strength of the staticmagnetic field, the frequency of transmitted radio-frequency energy canbe tuned to match the static magnetic field in the volume of interest.It can be appreciated that the NMR tool 10 may include a variety ofcomponents and configurations as known in the art of NMR. In that NMRtools are known in the art, specific details of components andconfigurations of these tools are not discussed in further detail.

The NMR tool 10 is conveyed through the borehole 2 by a carrier 5, whichcan be a drill tubular such as a drill string 6. A drill bit 7 isdisposed at the distal end of the drill string 6. A drill rig 8 isconfigured to conduct drilling operations such as rotating the drillstring 6 and thus the drill bit 7 in order to drill the borehole 2. Inaddition, the drill rig 8 is configured to pump drilling mud (i.e.,drill fluid) through the drill string 6 in order to lubricate the drillbit 7 and flush cuttings from the borehole 2. Downhole electronics 11are configured to operate the NMR tool 10, process measurement dataobtained downhole, and/or act as an interface with telemetry tocommunicate data or commands between downhole components and a computerprocessing system 12 disposed at the surface of the earth 3.Non-limiting embodiments of the telemetry include pulsed-mud and wireddrill pipe for real time communications. System operation and dataprocessing operations may be performed by the downhole electronics 11,the computer processing system 12, or a combination thereof. In analternative embodiment, the carrier 5 may be an armored wireline, whichmay also provide communications with the surface processing system 12.

Next, T2 inversions that combine long echo trains and trainlets arediscussed. There are several known methods of inversion that may be usedto achieve this. See for example variants of Separate Inversion (SI)(also called splicing technique) and Joint Inversion (JI) (also calledcomposite-data processing) in A METHOD FOR INVERTING NMR DATA SETS WITHDIFFERENT SIGNAL TO NOISE RATIOS, K. J. Dunn, D. J. Bergman, G. A.LaTorraca, S. M. Stonard, and M. B. Crowe; SPWLA 39^(th) Annual LoggingSymposium, May 26-29, 1998. That paper is called REF1 in thisdisclosure.

In the joint inversion (JI) technique, the multi-exponentialapproximation equations for the measured data may be represented as:

$\begin{matrix}{{{EET}_{i} = {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{i}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{ET}}{{R \cdot T}\; 2_{k}}}} )} )}}{{ETL}_{j} = {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{j}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{TL}}{{R \cdot T}\; 2_{k}}}} )} )}}} & (1)\end{matrix}$where EET_(i) is the i^(th) echo amplitude at time t_(i) of the longecho train with a long wait time TW_(ET), and ETL_(j) is the j^(th) echoamplitude at time t_(j) of a trainlet with a short wait time TW_(TL).φ_(k)'s are the sought-after T2 components of the T2 distribution i.e.,the amplitudes of the exponential functions associated with chosen fixedT2_(k) (or T2_(k) bins—selected intervals into which T2's arecategorized) where k runs from 1 to the chosen number of T2 bins. Theφ_(k)'s are optimized during the inversion process to achieve the bestfit to the measured NMR data. The range of i runs from 1 to the numberof echoes of the long echo train, while the range of j runs from 1 tothe number of trainlet echoes. Ideally, TW_(ET) should be long enough(e.g. >5*T₁), where T₁ is the longest T₁ component of the formation, topolarize all NMR components fully in which case the term

$( {1 - {\mathbb{e}}^{- \frac{{TW}_{ET}}{{R \cdot T}\; 2_{k}}}} )$in the equation for EET_(i) is 1 and can be omitted. The parameter Rrequires optimization in the JI routine. It is a measure of the T1/T2 ofthe formation (and in addition, as will be seen later, can be a motionartifact detector). In other words, R is not calculated directly asT1/T2, but rather is fitted to the echo train data by being optimized bythe JI routine. Echo trains and trainlets may have different interechotimes TE. The long echo train and averaged trainlets need weightingaccording to their number of averages, which is equivalent to thesquares of their inverted measurement errors (see REF1).

In separate inversion (SI), the averaged echo trains are inverted (i.e.,multiexponential fit) giving an echo train T2 distribution. The averagedtrainlets are inverted (separate from the echo trains, hence the nameSI) giving a trainlet T2 distribution. The principle for producing thefinal T2 distribution is by replacing in the echo train T2 distributionthe short-T2 components by the short-T2 components of the trainlet T2distribution. REF1 describes the details, including small modificationsto improve the accuracy e.g. by the method of Chen and Georgi 1997.

Next, motion artifacts are discussed. Movement of an NMR tool, duringthe sampling or receiving of the NMR echoes, might cause motionartifacts in the decay of the long echo train and to a smaller degree inthe trainlets. The main motion artifact is a reduction of FF componentsin the T2 distribution. What is lost from the FF component is thenmainly found in an increase of the BW component. Looking alone at thelong echo train, it is not possible to decide whether the BW component(or part of it) is really BW or a motion artifact.

When inverting long echo trains with long wait time combined with thetrainlets with short wait time using the prior art inversion methods(such as REF1), the fitted T2 decays either do not fit perfectly to bothtypes of echo trains, or R=T1/T2 is fitted unrealistically high if bothecho train types were subjected to the same motion. As disclosed herein,this misfit is used to detect and correct motion artifacts in the NMRdata. The disclosed inversion methods can find motion artifacts andremove or at least reduce them resulting in more accurate NMR data andmore accurate formation property values derived from the corrected NMRdata.

Next, manifestations of motion artifacts in NMR echo trains arediscussed with reference to FIG. 2. Such a motion artifact asillustrated in FIG. 2 can only occur if free fluid (FF) is present inthe formation, when some of the fractional porosity of the FF isconverted by the motion to a fractional porosity with apparent T2 in theregion of BW. FIG. 2 illustrates these ideas. In FIG. 2, the horizontalaxis is the time axis[s] and the vertical axis is the NMR amplitude. Ameasured NMR echo decay might look like trace FFBW_ET (except for thelack of noise). From this echo train alone, it cannot be decided whetherthe fast decay at the beginning of that echo train is due to bound water(BW) content of 20% and the remainder a free fluid (FF) content of 80%or whether the BW is not real and in fact a motion artifact. In thelatter case, the real FF would be 100% (trace FF_ET). It can be seenthat the appearance of the trainlets allows motion artifacts to bedistinguished from true bound water. Observe the two trainlet traces inthe bottom left of the diagram for which the wait time (i.e.polarization time) is short so that the FF component is polarized to asmall degree only. Trace FFBW_TL shows how the trainlet would look ifthe NMR sample was composed of 80% free fluid (FF) and 20% BW. The FFcontent is nearly fully suppressed by the short TW while the BW contentis preserved with only a little attenuation. Trace FFmot_TL shows howthe trainlet would look if the NMR sample was composed of 100% FF andthe fast decay at the beginning of trace FFBW_ET was a motion artifact.In this case, the trainlet trace would be, as shown, a scaled-downversion of the early times of trace FFBW_ET. Hence, by comparing theshape of the trainlet to the shape of the corresponding beginningportion of the long echo train, it can be determined if that portion isdue to BW or a motion artifact. In one or more embodiments, BW and amotion artifact may be present at the same time. In such a case it maybe challenging to identify by eye a motion artifact. Instead analgorithm may be used as described further down in this description.

Fortunately, it does not matter much whether the motion that was presentduring the long-TW echo train is or is not present during the trainlets.This is because the trainlets anyway show very little motion artifactsas the motion artifacts considered here can only be generated in thepresence of FF NMR signals; yet FF NMR signals are very much suppressedin the trainlets with their short TW.

This concept is applicable if the motion artifact in the long-TW echotrain looks similar to a fast decaying bound water component. This isgenerally the case if the artifact is caused by eccentric rotation of anaxisymmetric NMR tool or by centric rotation when its magnetic field isnot perfectly axi-symmetric. If the motion starts later in the echotrain (such as by a sudden shock), the motion artifact is less of aproblem because it is less likely to be mistaken for a wrong BWcomponent.

Next, joint inversion with motion artifact correction (JIMC) isdiscussed. The two fit-able equations (1) presented above are modifiedwith an additional multiplicative term, fitting the motion effect. Apossible term is

$1 - {A\;{{mot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t}{Tmot}}} )}}$with Amot and Tmot being the severeness (amplitude) and characteristictransient time of the motion artifact, respectively. The complete set offitable equations for the T2 inversion is now:

$\begin{matrix}{\mspace{79mu}{{{EET}_{i} = {( {1 - {A\;{{mot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t_{i}}{T\;{mot}}}} )}}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{i}}{T\; 2_{k}}}} )}}}{{ETL}_{j} = {( {1 - {A\;{{mot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t_{j}}{T\;{mot}}}} )}}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{j}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{TL}}{{R \cdot T}\; 2_{k}}}} )} )}}}}} & (2)\end{matrix}$where R=T1/T2 requires optimization in the joint inversion routine andthe long echo train and averaged trainlets need weighting according totheir number of averages which is equivalent to the squares of theirinverted measurement errors (see REF1). In the first equation it isassumed that the wait time TW_(ET) is long enough to polarize the NMRnuclei fully. If this was not the case, then an appropriate recoveryterm needs adding as in equations (1). The new parameters are Amot andTmot. The unknowns are: φ_(k), R=T1/T2, Amot and Tmot. It is useful toconstrain these fitable parameters to real values ≧0. This may be done,for example, by substituting the squares of the square roots of theseparameters in the equations before the fitting or by some other means.In one or more embodiments, R is constrained to be greater than one andAmot is constrained to be between zero and one.

In the above fitting equations (2), it is assumed that the same motionis present during the long-TW echo train and the short-TW trainlet. Inreality, this is not necessarily the case but should be of minorconsequence because the trainlets show very little FF signal andtherefore in the trainlets not much FF can be converted to BW by themotion.

Next, joint inversion with motion correction using a long echo train andtwo trainlet types is discussed. The system of equations (2) can beextended for one echo train with full polarization and two trainletswith different wait times TW_(TL1) and TW_(TL2). The complete set offitable equations for the T2 inversion then becomes:

$\begin{matrix}{\mspace{79mu}{{{EET}_{i} = {( {1 - {{Amot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t_{i}}{T\;{mot}}}} )}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{i}}{T\; 2_{k}}}} )}}}\text{}{{{ETL}\; 1_{j\; 1}} = {( {1 - {{Amot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t_{j\; 1}}{Tmot}}} )}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{j\; 1}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{{TL}\; 1}}{{R \cdot T}\; 2_{k}}}} )} )}}}{{{ETL}\; 2_{j\; 2}} = {( {1 - {{Amot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t_{j\; 2}}{Tmot}}} )}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{j\; 2}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{{TL}\; 2}}{{R \cdot T}\; 2_{k}}}} )} )}}}}} & (3)\end{matrix}$It can be appreciated that this system of equations can be extended toany number of echo trains and trainlets. Trainlets with medium wait timeTW (i.e., greater than short TW and less than long TW), or with twodifferent TWs, are in particular useful for a realistic determination ofthe fitting parameter R, noting that a realistic determination of R isthe precondition for obtaining a realistic fitting of the motionartifact.

It should be noted that the different echo trains of equations (2) and(3) do not need to have the same interecho time TE. These equations canbe used with echo trains with different numbers of echoes NE anddifferent interecho times TE.

Next, examples of applying the joint inversion with motion artifactcorrection are presented. These examples are based on simulated NMR echodecays with simulated motion artifacts. For this section, the parametersare: FF are T2 components greater than 100 ms, BW are T2 componentsbetween 3.3 millisecond (ms) and 100 ms, and CBW are T2 components lessthan 3.3 ms.

A free fluid (FF) example is presented with one relaxation component ofT2=1 sec. and R=T1/T2=1.5. The long echo train is fully polarized whilethe trainlets use a wait time of 60 ms and are weighted 96 times with√{square root over (96)} times lower noise. The prior art jointinversion (JI) is compared to the invented joint inversion with motioncorrection (JIMC).

FIG. 3 depicts aspects of an NMR echo train and trainlet used todemonstrate prior art joint inversion in the presence of a motionartifact. FIG. 3A illustrates a noisy simulated echo decay of 1000echoes (ET) together with a trainlet (bottom left corner). Thehorizontal axis is the time axis with time in seconds. The vertical axisis the echo amplitude axis, normalized to 100% for the start of an echotrain without artifact and noise and with full polarization. The dottedtrace indicates an artifact-free mono-exponential decay with acharacteristic decay time of 1 second—the expected noise free and motionartifact free echo decay. The deviation between noisy trace and thedotted trace is due to noise and motion artifact. The prior art jointinversion obtains the fits in the long echo train (trace FET) and in thetrainlet (trace FTL). FIG. 3B shows FIG. 3A, but with the time axismagnified (i.e., zoomed in) to show the times 0 sec to 0.05 sec only.FIG. 3C is also a magnified view of FIG. 3A, but showing details of thetrainlet TL with its fit FTL. It is seen in FIG. 3B that the fit (FET)of the inversion is systematically too low at the start of ET (whichresults in too low total porosity). Similarly, in FIG. 3C the fit FTLshows a systematic deviation (i.e., misfit) from the trainlet TL (i.e.,wrong slope). (A motion corrected fit should find the true motionartifact-free echo decay ETorg.) The T2 distribution obtained by theprior art joint inversion is given in FIG. 4 in which the horizontaltime bin axis is in seconds. The distribution of Φ_(k) in FIG. 4provides a determination of certain properties:

Total porosity: 95.1% (true 100%);

FF: 88.0% (true 100%);

BW: 7.1% (true 0%);

CBW: 0.0% (true 0%); and

R=T1/T2: 3.2 (true 1.5).

Both, the R fitted unrealistically high, and the misfits in FIGS. 3B and3C are indications of an incompatibility of the trainlets TL with theecho train ET with regards to data processing that does not comprisemotion artifacts. While a misfit (systematic deviation from the data tobe fitted) is either obvious or can be determined by an appropriatealgorithm, an R fitted too high can be identified by comparing with alikely R for a particular formation type. (A JI with motion correction,i.e. JIMC, should find an R nearer to the true R.) A likely or referenceR for a formation type may be determined by testing of an NMR tool indifferent formation types or by analysis so that the fitted R can becompared to the likely R. Hence, by knowing if the fitted R for a typeof formation being drilled differs from the likely R by more than athreshold value, then an indication can be provided that the fitted R isincompatible with the NMR data.

FIG. 5 is similar to FIG. 3 and illustrates a noisy long echo traintrace (ET) and a fit to this trace (FET) using JIMC. FIGS. 5B and 5C aremagnified versions of FIG. 5A (in the same way that FIGS. 3B and 3C weremagnified versions of FIG. 3A). The FET trace reproduces the totalporosity at the start of the long echo train correctly (FIG. 5B) and thefit of the trainlet, FIG. 5C, is better than that of FIG. 3C. Theresulting T2 distribution using JIMC is illustrated in FIG. 6. If themulti-exponential decay, generated from the above mentioned T2distribution in FIG. 6, is plotted, the trace (ETcor) just below thedotted trace (ETorg) in FIGS. 5A and 5B is obtained. This is the motioncorrected fit, which is not far from the artifact-free dotted trace,ETorg. The differences to the true T2 components are very small. Themotion artifact is almost completely removed. The FF component with itstwo peaks in the T2 distribution may look irregular. This is because theJIMC of the equations (2) does not yet include a proper regularization.In one or more embodiments, such regularization would be used. Thedistribution of Φ_(k) in FIG. 6 provides a determination of certainproperties:

Total porosity: 99.9% (true 100%);

FF: 99.7% (true 100%);

BW: 0.2% (true 0%);

CBW: 0.0% (true 0%);

R=T1/T2: 1.9 (true 1.5);

Amot: 0.13; and

Tmot: 0.033 s.

The estimates of these properties using JIMC are more accurate than theestimates using prior art JI in the above paragraph.

Another example using both JI and JIMC is presented with respect tocharacterizing shaly sand. This is a simulated example with threerelaxation components: 60% FF with T2=1 s, 30% BW with T2=15 ms, 10% CBWwith T2=1.5 ms and R=T1/T2=1.5 for all components. The long echo trainis fully polarized while the trainlets use a wait time of 60 ms and areweighted 96 times with √{square root over (96)} times lower noise. Theexample further compares the output of JIMC using one or two trainlets.The figures used in this example are equivalent to the figures used inthe free fluid example. FIGS. 7 and 8 relate to using the prior art JIwith a long echo train and one trainlet with wait time TW=60 ms. Thedistribution of Φ_(k) in FIG. 8 provides a determination of certainproperties:

Total porosity: 102.0% (true 100%); FF: 51.6% (true 60%); BW: 40.6%(true 30%); CBW: 9.9% (true 10%); and R = T1/T2: 2.7 (true 1.5).

FIGS. 9 and 10 relate to using JIMC with a long echo train and onetrainlet with wait time TW=60 ms. The distribution of Φ_(k) in FIG. 10provides a determination of certain properties:

Total porosity: 102.0% (true 100%); FF: 51.8% (true 60%); BW: 40.2%(true 30%); CBW: 9.9% (true 10%); R = T1/T2: 2.6 (true 1.5); Amot:0.000004; and Tmot: 0.131 s.It is noted that in this example the accuracy using JIMC is comparableto the accuracy using the prior art JI (Amot is fitted to almost zero).However, a good fit is found with an excessive R=2.6. Because of thegood fit, it is clear that not enough information for motion correctionis contained in the long echo train and the trainlet for this example.

Because in the previous example there was not enough information in thelong echo train and the trainlet, a second trainlet is added with a waittime TW=1 sec that is different to that of the first trainlet (TW=60ms). FIGS. 11 and 12 relate to using JIMC with a long echo train, onetrainlet with wait time TW=60 ms, and one trainlet with wait time TW=1s. The distribution of Φ_(k) in FIG. 12 provides a determination ofcertain properties:

Total porosity: 101.0% (true 100%); FF: 60.4% (true 60%); BW: 29.7%(true 30%); CBW: 10.9% (true 10%); R = T1/T2: 1.4 (true 1.5); Amot:0.16; and Tmot: 0.041 s.In this example with two trainlets, the JIMC finds the correct R and themotion artifact and correctly removes the artifact. Further, looking atFIG. 11, it is seen that all fits derived from the T2 distribution ofFIG. 12 and plotted in FIG. 11 (FET, FTL1, FTL2) are excellent. The FETtrace in FIG. 11 is the result of the fitted equations (3) and fits theecho train with its artifacts. The ETcor trace in FIG. 11 is thecorrected echo train, using the fitted T2 distribution and R in thefirst equation (EET) of equations (3) while leaving off the motion term,

$1 - {{Amot} \cdot {( {1 - {\mathbb{e}}^{- \frac{t_{i}}{Tmot}}} ).}}$The corrected echo train reproduces faithfully the original echo trainwithout motion artifact or noise.

FIG. 13 presents tabled results of the above-presented examples. Thelast line shows that JIMC with two trainlets obtains approximately thecorrect T2 components, even in the more difficult case of Shaly Sandwith the following properties: Total porosity=100%; CBW (0 to 3.3ms)=10%; BW (3.3. to 100 ms)=30%; and FF (0.1 to 4 sec)=60%.

Embodiments of motion artifact correction, described in the foregoingdescription, are variants of the disclosed Joint Inversion with Motionartifact Correction (JIMC), which are modifications of the prior artJoint Inversion (JI). It is to be understood, though, that otherembodiments, like variants of Separate Inversion with Motion artifactCorrection (SIMC), which are modifications of the prior art SeparateInversion (SI), are well within the scope of this patent application.

Aspects of the process of correcting for a motion artifact when SI isused for inversion of T2 data (SIMC) are now discussed. In SIMC, theaveraged echo trains are inverted giving an echo train T2 distribution.(It is assumed that there is a motion artifact in the echo train, whichcaused a too high amplitude of the short-T2 components and a too lowamplitude of the long-T2 components.) Now a combined T2 distribution isproduced by replacing in the echo train T2 distribution the short T2components by the short T2 components of the trainlet T2 distribution(so far identical to SI). This combining reduces the motion artifact inthe replaced short-T2 components but the long-T2 components have stilltoo low amplitude and therefore the total porosity is too low, too. Toreduce now the motion artifact in the long-T2 components of thedistribution, the short-T2 components of the trainlet are summed andthis sum subtracted from the summed short-T2 components of the echotrain, resulting in a difference. Then, this difference is distributedacross the long T2 components of the combined T2 distribution. Thisresults in getting a final T2 distribution with motion-artifact-reducedshort and long T2 components and more correct total porosity. The resultmay be improved by using the method of Chen and Georgi 1997 as describedin REF1.

In the prior art SI the TW of the trainlets is often set quite short,just to polarize the CBW but not the BW. As very often the motioneffects manifest themselves in the BW region, the SIMC, as describedabove, may not be efficient in reducing motion artifacts. Rather, neededin addition to the CBW trainlets with a TW just long enough to polarizeCBW, are also BW trainlets with a TW long enough to polarize the BWcomponents. The CBW trainlets are then used as in prior art SI while theBW trainlets are used for the motion artifact correction. The sequenceof processing will then be: produce a combined T2 distribution byreplacing in the echo train T2 distribution the CBW-T2 components by theCBW-T2 components of the CBW-trainlet T2 distribution and the BW-T2components by the BW components of the BW-trainlet T2 distribution. If amotion artifact was present in the replaced BW-T2 components of the echotrain, then this motion artifact is now reduced in the combined T2distribution but the long T2 components have still too low amplitude andtherefore the total porosity is too low, too. To reduce now the motionartifact in the long-T2 components of the distribution: the BW-T2components of the BW trainlet are summed and this sum subtracted fromthe summed BW-T2 components of the echo train, resulting in adifference. Then this difference is distributed across the long T2components of the combined T2 distribution. This results in getting afinal T2 distribution with motion-artifact-reduced short and long T2components and more correct total porosity. The result may again beimproved by using the method of Chen and Georgi 1997 as described inREF1. It can be appreciated that some motion artifacts may not affectthe BW T2 components but may affect the CBW T2 components or both. T2components affected by motion artifacts in the CBW region may becorrected using techniques similar to the above described techniques forcorrecting motion artifacts in the BW region.

FIG. 14 is a flow chart for a method 140 for processing nuclear magneticresonance (NMR) measurement data. Block 141 calls for receiving, with aprocessor, NMR measurement data obtained from an NMR tool, the NMRmeasurement data being affected by a motion artifact and comprising afirst echo train obtained with a long polarization time TW_(ET) and asecond echo train obtained with a short polarization time TW_(TL) thatis shorter than TW_(ET). Block 142 calls for at least one of (i)reducing, with a processor, an effect on the NMR measurement data of themotion artifact using the first echo train and the second echo train and(ii) identifying, with a processor, the motion artifact using the firstecho train and the second echo train, wherein the motion artifact isrelated to a magnetic field magnitude that varies in a volume ofinterest due to a motion of the NMR tool. The motion may be due to atleast one of radial movement, axial vibration, and rotation withnon-axial symmetry of magnetic fields of the NMR tool. The NMR tool maybe conveyed through a borehole penetrating an earth formation by acarrier such as a drill tubular.

Regarding the reducing in the method 140, reducing may include using acorrecting inversion method that models the motion artifact to provide acorrected transverse relaxation time constant (T2) distribution. Thecorrecting inversion may include using the following multiplicativeterm:

$1 - {{Amot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t}{Tmot}}} )}$where Amot represents an amplitude of the motion artifact, Tmotrepresents a transient time constant of the motion artifact, and trepresents time. The reducing may include using the equations (2) whenthe second echo train is a single echo train and equations (3) when thesecond echo train includes two echo trains.

Regarding the identifying in the method 140, identifying may includedetermining if multi-exponential approximations of T2 distributions ofthe first echo train and the second echo train obtained using anon-correcting inversion method that does not model the motion artifactprovide an indication of incompatibility between the long echo train andthe short echo train. The indication of incompatibility may bedetermined by a user who reviews the multi-exponential approximationsand inputs into the identifying processor that there is an indication ofincompatibility. In one example, the indication of incompatibility mayexist when R is unreasonably high based on the details of the earthformation of interest. Equations (1) may be used for themulti-exponential approximations. Identifying may also include executingan algorithm that provides the indication of incompatibility. In oneexample of the algorithm, the algorithm may include equations (1). Amotion artifact is indicated if either the joint fit of the twoequations (1) is bad or the fitted R is excessively high. In anotherexample of the algorithm, the algorithm may include equations (4).

$\begin{matrix}{{{EET}_{i} = {\sum\limits_{k}( {\phi\;{E_{k} \cdot {\mathbb{e}}^{- \frac{t_{i}}{T\; 2_{k}}}}} )}}{{ETL}_{j} = {\sum\limits_{k}( {\phi\;{T_{k} \cdot {\mathbb{e}}^{- \frac{t_{j}}{T\; 2_{k}}}}} )}}} & (4)\end{matrix}$For equations (4) when acquiring the EET_(i) the polarization time ischosen to be long enough to substantially polarize FF and when acquiringthe ETL_(j) the polarization time is chosen to be long enough tosubstantially polarize BW but not FF. A motion artifact is then detectedwhen the sum of those φE_(k) that depend on BW is substantially greaterthan (e.g., greater by more than 10%) the sum of those φT_(k) thatdepend on BW. The φE_(k) and φT_(k) that depend on BW are a subset ofall the φE_(k) and φT_(k).

The method 140 may also include providing a corrected T2 distribution byreducing the effect of the motion artifact and then estimating aproperty of the earth formation using the corrected T2 distribution.

Motion artifact detection as well as motion artifact correction may beperformed either downhole or uphole preferably in real time, or upholewhen post processing the NMR data.

In support of the teachings herein, various analysis components may beused, including a digital and/or an analog system. For example, thedownhole electronics 11, the computer processing system 12, or the NMRtool 10 may include digital and/or analog systems. The system may havecomponents such as a processor, storage media, memory, input, output,communications link (wired, wireless, pulsed mud, optical or other),user interfaces, software programs, signal processors (digital oranalog) and other such components (such as resistors, capacitors,inductors and others) to provide for operation and analyses of theapparatus and methods disclosed herein in any of several mannerswell-appreciated in the art. It is considered that these teachings maybe, but need not be, implemented in conjunction with a set of computerexecutable instructions stored on a non-transitory computer readablemedium, including memory (ROMs, RAMs), optical (CD-ROMs), or magnetic(disks, hard drives), or any other type that when executed causes acomputer to implement the method of the present invention. Theseinstructions may provide for equipment operation, control, datacollection and analysis and other functions deemed relevant by a systemdesigner, owner, user or other such personnel, in addition to thefunctions described in this disclosure.

Further, various other components may be included and called upon forproviding for aspects of the teachings herein. For example, a powersupply (e.g., at least one of a generator, a remote supply and abattery), cooling component, heating component, magnet, electromagnet,sensor, electrode, transmitter, receiver, transceiver, antenna,controller, optical unit, electrical unit or electromechanical unit maybe included in support of the various aspects discussed herein or insupport of other functions beyond this disclosure.

The term “carrier” as used herein means any device, device component,combination of devices, media and/or member that may be used to convey,house, support or otherwise facilitate the use of another device, devicecomponent, combination of devices, media and/or member. Other exemplarynon-limiting carriers include drill strings of the coiled tube type, ofthe jointed pipe type and any combination or portion thereof. Othercarrier examples include casing pipes, wirelines, wireline sondes,slickline sondes, drop shots, bottom-hole-assemblies, drill stringinserts, modules, internal housings and substrate portions thereof.

The flow diagrams depicted herein are just examples. There may be manyvariations to these diagrams or the steps (or operations) describedtherein without departing from the spirit of the invention. Forinstance, the steps may be performed in a differing order, or steps maybe added, deleted or modified. All of these variations are considered apart of the claimed invention.

Elements of the embodiments have been introduced with either thearticles “a” or “an.” The articles are intended to mean that there areone or more of the elements. The terms “including” and “having” areintended to be inclusive such that there may be additional elementsother than the elements listed. The conjunction “or” when used with alist of at least two terms is intended to mean any term or combinationof terms. The terms “first,” “second” and the like do not denote aparticular order, but are used to distinguish different elements.

While one or more embodiments have been shown and described,modifications and substitutions may be made thereto without departingfrom the spirit and scope of the invention. Accordingly, it is to beunderstood that the present invention has been described by way ofillustrations and not limitation.

It will be recognized that the various components or technologies mayprovide certain necessary or beneficial functionality or features.Accordingly, these functions and features as may be needed in support ofthe appended claims and variations thereof, are recognized as beinginherently included as a part of the teachings herein and a part of theinvention disclosed.

While the invention has been described with reference to exemplaryembodiments, it will be understood that various changes may be made andequivalents may be substituted for elements thereof without departingfrom the scope of the invention. In addition, many modifications will beappreciated to adapt a particular instrument, situation or material tothe teachings of the invention without departing from the essentialscope thereof. Therefore, it is intended that the invention not belimited to the particular embodiment disclosed as the best modecontemplated for carrying out this invention, but that the inventionwill include all embodiments falling within the scope of the appendedclaims.

What is claimed is:
 1. A method for processing nuclear magneticresonance (NMR) measurement data, the method comprising: conveying adownhole NMR tool through a borehole penetrating the earth and operatingthe downhole NMR tool in the borehole, the downhole NMR tool comprisinga processor; receiving, with the processor, NMR measurement dataobtained from the downhole NMR tool, the NMR measurement data beingaffected by a motion artifact and comprising a first echo train obtainedwith a long polarization time TW_(ET) and a second echo train obtainedwith a short polarization time TW_(TL) that is shorter than TW_(ET);providing, with the processor, an indication of incompatibility betweenthe first echo train and the second echo train; at least one of (i)reducing, with the processor, an effect on the NMR measurement data ofthe motion artifact using the first echo train and the second echo trainusing the indication of incompatibility and (ii) identifying, with theprocessor, the motion artifact using the first echo train and the secondecho train using the indication of incompatibility; and applying motioncorrection, with the processor, on the NMR measurement data using theindication of compatibility; wherein the motion artifact is related to amagnetic field magnitude that varies in a volume of interest due to amotion of the NMR tool.
 2. The method according to claim 1, wherein themotion of the NMR tool comprises at least one of radial movement, axialvibration, and rotation with non-axial symmetry of magnetic fields ofthe NMR tool.
 3. The method according to claim 1, wherein reducingcomprises using a correcting inversion method that models the motionartifact to provide a corrected transverse relaxation time constant (T2)distribution.
 4. The method according to claim 3, wherein the correctinginversion method comprises the following multiplicative term:$1 - {{Amot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t}{Tmot}}} )}$where Amot represents an amplitude of the motion artifact, Tmotrepresents a transient time constant of the motion artifact, and trepresents time.
 5. The method according to claim 4, wherein the secondecho train comprises one echo train and the correcting inversion methodcomprises the following equations:${EET}_{i} = {( {1 - {{Amot}( {1 - {\mathbb{e}}^{- \frac{t_{i}}{Tmot}}} )}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{i}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{ET}}{{R \cdot T}\; 2_{k}}}} )} )}}$${ETL}_{j} = {( {1 - {{Amot}( {1 - {\mathbb{e}}^{- \frac{t_{j}}{Tmot}}} )}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{j}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{ET}}{{R \cdot T}\; 2_{k}}}} )} )}}$where EET_(i) is the i^(th) echo amplitude at time t_(i) of the firstecho train with a long wait time TW_(ET), and ETL_(j) is the j^(th) echoamplitude at time t_(j) of the second echo train with a short wait timeTW_(TL), φ_(k) represent amplitudes of exponential functions associatedwith chosen fixed T2_(k) time interval bins with k running from 1 to achosen number of T2 time interval bins, T2_(k) representing a T2 time ofthe T2_(k) time bin, and R represents T1/T2.
 6. The method according toclaim 5, wherein the term$( {1 - {\mathbb{e}}^{- \frac{{TW}_{ET}}{{R \cdot T}\; 2_{k}}}} )$is estimated to be one.
 7. The method according to claim 4, wherein thesecond echo train comprises two echo trains, ETL1 having wait timeTW_(TL1) and ETL2 having wait time TW_(TL2), each of the second echotrains having a different wait time and the correcting inversion methodcomprises the following equations:$\mspace{79mu}{{EET}_{i} = {( {1 - {{Amot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t_{i}}{T\;{mot}}}} )}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{i}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{ET}}{{R \cdot T}\; 2_{k}}}} )} )}}}$${{ETL}\; 1_{j\; 1}} = {{( {1 - {{Amot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t_{j\; 1}}{Tmot}}} )}} ) \cdot {\sum\limits_{k}{( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{j\; 1}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{{TL}\; 1}}{{R \cdot T}\; 2_{k}}}} )} ){ETL}\; 2_{j\; 2}}}} = {( {1 - {{Amot} \cdot ( {1 - {\mathbb{e}}^{- \frac{t_{j\; 2}}{Tmot}}} )}} ) \cdot {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{j\; 2}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{{TL}\; 2}}{{R \cdot T}\; 2_{k}}}} )} )}}}$where EET_(i) is the i^(th) echo amplitude at time t_(i) of the firstecho train with a long wait time TW_(ET), ETL1_(j1) is the j1^(th) echoamplitude at time t_(j1) of one of second echo trains with a short waittime TW_(TL1), ETL2_(j2) is the j2^(th) echo amplitude at time t_(j2) ofthe other second echo train with a short wait time TW_(TL2), φ_(k)represent amplitudes of exponential functions associated with chosenfixed T2_(k) time interval bins with k running from 1 to a chosen numberof T2 time interval bins, T2_(k) representing a T2 time of the T2_(k)time bin, and R represents T1/T2.
 8. The method according to claim 7,wherein the term$( {1 - {\mathbb{e}}^{- \frac{{TW}_{ET}}{{R \cdot T}\; 2_{k}}}} )$is estimated to be one.
 9. The method according to claim 1, whereincorrecting comprises using Separate Inversion with Motion artifactCorrection (SIMC).
 10. The method according to claim 1, whereinidentifying comprises determining if multi-exponential approximations ofT2 distributions of the first echo train and the second echo trainobtained using a non-correcting inversion method that does not model themotion artifact provide an indication of incompatibility between thelong echo train and the short echo train.
 11. The method according toclaim 10, wherein the indication of incompatibility is input into theidentifying processor by a user upon reviewing the multi-exponentialapproximations.
 12. The method according to claim 11, wherein the userprovides the indication of incompatibility when R exceeds a thresholdvalue.
 13. The method according to claim 11, further comprisinginputting into the reducing processor instructions to use a correctinginversion method that models the motion artifact to provide a correctedtransverse relaxation time constant (T2) distribution.
 14. The methodaccording to claim 10, wherein the multi-exponential approximationsusing the non-correcting inversion method are determined by using thefollowing equations:${EET}_{i} = {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{i}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{ET}}{{R \cdot T}\; 2_{k}}}} )} )}$${ETL}_{j} = {\sum\limits_{k}( {\phi_{k} \cdot {\mathbb{e}}^{- \frac{t_{j}}{T\; 2_{k}}} \cdot ( {1 - {\mathbb{e}}^{- \frac{{TW}_{TL}}{{R \cdot T}\; 2_{k}}}} )} )}$where EET_(i) is the i^(th) echo amplitude at time t_(i) of the firstecho train with a long wait time TW_(ET), ETL_(j) is the j^(th) echoamplitude at time t_(j) of the second echo train with a short wait timeTW_(TL), φ_(k) represent amplitudes of exponential functions associatedwith chosen fixed T2_(k) time interval bins with k running from 1 to achosen number of T2 time interval bins, T2_(k) representing a T2 time ofthe T2_(k) time bin, and R represents T1/T2.
 15. The method according toclaim 14, wherein the indication of incompatibility is input into theidentifying processor by a user upon reviewing the multi-exponentialapproximations and the user provides the indication of incompatibilitywhen R exceeds a threshold value for a type of formation being logged orhas a fit of the NMR data that exceeds a quantitative fit error value.16. The method according to claim 10, further comprising executing analgorithm that provides the indication of incompatibility.
 17. Themethod according to claim 16, wherein the algorithm comprises thefollowing equations for the non-correcting inversion method:${EET}_{i} = {\sum\limits_{k}( {\phi\;{E_{k} \cdot {\mathbb{e}}^{- \frac{t_{i}}{T\; 2_{k}}}}} )}$${ETL}_{j} = {\sum\limits_{k}( {\phi\;{T_{k} \cdot {\mathbb{e}}^{- \frac{t_{j}}{T\; 2_{k}}}}} )}$where EET_(i) is the i^(th) echo amplitude at time t_(i) of the firstecho train with a long wait time TW_(ET), ETL_(j) is the j^(th) echoamplitude at time t_(j) of the second echo train with a short wait timeTW_(TL), φE_(k) and φT_(k) represent amplitudes of exponential functionsassociated with chosen fixed T2_(k) time interval bins with k runningfrom 1 to a chosen number of T2 time interval bins, T2_(k) representinga T2 time of the T2_(k) time bin, and R represents T1/T2; and whereTW_(ET) is long enough to substantially polarize free fluid and TW_(TL)is long enough to substantially polarize bound water and short enough tosubstantially not polarize free fluid components of a formation.
 18. Themethod according to claim 17, wherein the algorithm provides theindication of incompatibility when the sums of those φE_(k) that dependon the bound water or clay bound water components are substantiallygreater than the sums of those φT_(k) that that depend on the boundwater or clay bound water components.
 19. A method for performingnuclear magnetic resonance (NMR) measurements on an earth formation, themethod comprising: conveying a downhole NMR tool through a boreholepenetrating the earth formation and operating the downhole NMR tool inthe borehole, the downhole NMR tool comprising a processor; receiving,with the processor, NMR measurement data obtained from the NMR tooldisposed on a carrier, the NMR measurement data being affected by amotion artifact and comprising a first echo train obtained with a longpolarization time TW_(ET) and a second echo train obtained with a shortpolarization time TW_(TL) that is shorter than TW_(ET); providing, withthe processor, an indication of incompatibility between the first echotrain and the second echo train; at least one of (i) reducing, with aprocessor, an effect on the NMR measurement data of the motion artifactusing the first echo train and the second echo train using theindication of incompatibility and (ii) identifying, with a processor,the motion artifact using the first echo train and the second echo trainusing the indication of incompatibility; applying motion correction,with the processor, on the NMR measurement data to generate motioncorrected NMR measurement data using the indication of compatibility;and estimating, with the processor, at least one of a T2 distribution, atotal porosity, a fractional porosity, a pore size, and a fluid typeusing the motion corrected NMR measurement data; wherein the motionartifact is related to a magnetic field magnitude that varies in avolume of interest due to a motion of the NMR tool.
 20. The methodaccording to claim 19, wherein the effect of the motion artifact isreduced to provide a corrected T2 distribution and the method furthercomprises estimating a property of the earth formation using thecorrected T2 distribution.
 21. The method according to claim 20, whereinthe NMR tool is coupled to a carrier configured to be conveyed throughthe borehole.
 22. The method according to claim 21, wherein the carrieris a drill tubular.